Is the energy conserved FOR EACH BODY in a two-body central force problem?

I would like to know if the energy of each body of a two body gravitationnal problem is separately conserved. I know that the individual angular momentum are separately conserved and that the TOTAL energy of the two bodies is conserved. However, I don't know if there could be energy transfer between the two bodies or if it's forbidden by symmetry considerations.

In my mechanics textbook, the two-body central force problem is only treated as a one-dimensional problem of the motion of a reduced mass in an effective potential, where only the energy of a reduced mass orbiting around the center of mass is considered. This confuses me when I try to think of the energy of ONE of the two bodies. Is the energy of ONE of the two bodies conserved? Is there a way to see this?

Philip is correct. As an example, think of a system of two point particles which only interact via gravity. The only relevant energies here are kinetic and potential. If we pick some given reference frame, then each body has some kinetic energy (which could be zero) but the potential energy is just the gravitational potential energy GMm/r. Which body does this potential energy belong to? The energy only exists as an interaction between the two bodies.

Imagine we had one particle very massive (and thus roughly stationary), and the other very light and in a highly elliptical orbit--let's call this a Sun-comet system. When the comet is very close to the Sun, it's moving very fast and thus has high kinetic energy, but when the comet is far from the Sun, it moves much slower. The comet's kinetic energy is not conserved. But the potential energy has changed, and thus the system's total energy is conserved. But the potential energy does not necessarily "belong" to the comet, so it doesn't make sense to talk about the comet's total energy.

When we study central force motion, we do what you said: move to the center of mass frame and consider the motion of a single particle (with a mass equal to the reduced mass) inside a time-independent potential field. The problem has been changed: now there is only a field and a particle. So we can just say that all the energy belongs to the particle. But if we go back to the two-particle system, it's ambiguous which particle the energy belongs to.